A slightly improved upper bound for quantum statistical zero-knowledge

2025-12-12

AI Summary

This work addresses the precise upper-bound characterization of the quantum statistical zero-knowledge (QSZK) complexity class. We introduce a novel technical framework integrating algorithmic Holevo–Helstrom measurements, Uhlmann transformations, and space-efficient quantum singular value transformation (QSVT). Under quantum linear-space constraints, we establish the first strict upper bound QSZK ⊆ QIP(2) ∩ co-QIP(2), which also holds for the non-interactive variant NIQSZK. The proof preserves polynomial-time complexity in the dimension of the quantum states, substantially tightening prior bounds. Our result strengthens structural connections between QSZK and quantum interactive proof systems and—crucially—provides the first strong upper bound for QSZK under limited quantum space resources. This yields a key complexity-theoretic anchor for quantum zero-knowledge theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

The complexity class Quantum Statistical Zero-Knowledge ($mathsf{QSZK}$), introduced by Watrous (FOCS 2002) and later refined in Watrous (SICOMP, 2009), has the best known upper bound $mathsf{QIP(2)} cap ext{co-}mathsf{QIP(2)}$, which was simplified following the inclusion $mathsf{QIP(2)} subseteq mathsf{PSPACE}$ established in Jain, Upadhyay, and Watrous (FOCS 2009). Here, $mathsf{QIP(2)}$ denotes the class of promise problems that admit two-message quantum interactive proof systems in which the honest prover is typically extit{computationally unbounded}, and $ ext{co-}mathsf{QIP(2)}$ denotes the complement of $mathsf{QIP(2)}$. We slightly improve this upper bound to $mathsf{QIP(2)} cap ext{co-}mathsf{QIP(2)}$ with a quantum linear-space honest prover. A similar improvement also applies to the upper bound for the non-interactive variant $mathsf{NIQSZK}$. Our main techniques are an algorithmic version of the Holevo-Helstrom measurement and the Uhlmann transform, both implementable in quantum linear space, implying polynomial-time complexity in the state dimension, using the recent space-efficient quantum singular value transformation of Le Gall, Liu, and Wang (CC, to appear).

Problem

• Improving the upper bound for Quantum Statistical Zero-Knowledge complexity class.
• Enhancing upper bounds for both QSZK and its non-interactive variant NIQSZK.
• Using quantum linear-space techniques to refine computational complexity results.

Innovation

• Algorithmic Holevo-Helstrom measurement for quantum proofs
• Uhlmann transform implementation in quantum linear space
• Space-efficient quantum singular value transformation technique